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that there exist irrationals \(a\) and \(b\) such that \(a^{b}\) is rational:
\(\sqrt{2}\) is irrational. Let \(z=\sqrt{2}^{\sqrt{2}}\). If \(z\) is rational, the claim is proved. If not, then \(z^{\sqrt{2}}=2\) proves the claim.

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